Arithmetic properties of Fredholm series for p-adic modular forms

Abstract

We study the relationship between recent conjectures on slopes of overconvergent p-adic modular forms "near the boundary" of p-adic weight space. We also prove in tame level 1 that the coefficients of the Fredholm series of the U_p operator never vanish modulo p, a phenomenon that fails at higher level. In higher level, we do check that infinitely many coefficients are non-zero modulo p using a modular interpretation of the mod p reduction of the Fredholm series recently discovered by Andreatta, Iovita and Pilloni.

Final version. Numbering in main body different different from previous version. To appear in Proc. Lon. Math. Soc. 25 pages, 7 tables

References in corpus (2)

• The eigencurve over the boundary of weight space
• Slopes of eigencurves over boundary disks

Cited by in corpus (11)

• Extended eigenvarieties for overconvergent cohomology
• The eigencurve over the boundary of weight space
• Slopes of modular forms and the ghost conjecture
• Slopes of modular forms and the ghost conjecture, II
• Slopes of modular forms and the ghost conjecture (unabridged version)
• Upper bounds for constant slope $p$-adic families of modular forms
• A remark on non-integral p-adic slopes for modular forms
• Slopes of overconvergent Hilbert modular forms
• Irreducible components of extended eigenvarieties and interpolating Langlands functoriality
• Generic Newton Slopes for Artin-Schreier-Witt Tower in two variables
• An explicit computation of the Hecke operator and the ghost conjecture